加速外梯度型方法——第二部分：余次单调性下的推广与次线性收敛速率
Accelerated Extragradient-Type Methods -- Part 2: Generalization and Sublinear Convergence Rates under Co-Hypomonotonicity
AI总结：
本文在余次单调性和 Lipschitz 连续性假设下，统一并推广了锚定外梯度法和 Nesterov 加速外梯度法，建立了 $\mathcal{O}(1/k)$ 及 $o(1/k)$ 的最后迭代收敛速率，并通过数值实验验证了理论结果。
摘要：
继本项目第一部分之后，本文全面研究了两类基于外梯度的方法：锚定外梯度法和 Nesterov 加速外梯度法，用于求解[非]线性包含问题（特别是方程），主要在 Lipschitz 连续性和余次单调性假设下进行。我们将一类用于单调包含问题的锚定外梯度方法统一并推广到更广泛的方案中，涵盖现有算法作为特例。我们为该一般框架建立了底层映射残差范数的 $\mathcal{O}(1/k)$ 最后迭代收敛速率，然后将其特化以获得具体实例的收敛保证，其中 $k$ 表示迭代计数器。我们将方法扩展到一类锚定 Tseng 前向-后向-前向分裂方法，以获得更广泛的算法类来求解余次单调包含问题。我们同样分析了该一般方案的 $\mathcal{O}(1/k)$ 最后迭代收敛速率，并将其特化以获得现有变体和新变体的收敛结果。我们将 Nesterov 加速外梯度法推广并统一到一个新的算法类中，该类涵盖现有方案作为特例，同时生成新的变体。对于这些方案，我们能够在余次单调性条件下证明残差范数的 $\mathcal{O}(1/k)$ 最后迭代收敛速率，涵盖一类非单调问题。我们提出另一类新颖的 Nesterov 加速外梯度方法来求解包含问题。有趣的是，这些算法同时实现了 $\mathcal{O}(1/k)$ 和 $o(1/k)$ 的最后迭代收敛速率，并且在余次单调性和 Lipschitz 连续性条件下实现了迭代序列的收敛。最后，我们提供了一组涵盖不同场景的数值实验，以验证我们的算法和理论保证。
英文摘要：
Following the first part of our project, this paper comprehensively studies two types of extragradient-based methods: anchored extragradient and Nesterov's accelerated extragradient for solving [non]linear inclusions (and, in particular, equations), primarily under the Lipschitz continuity and the co-hypomonotonicity assumptions. We unify and generalize a class of anchored extragradient methods for monotone inclusions to a wider range of schemes encompassing existing algorithms as special cases. We establish $\mathcal{O}(1/k)$ last-iterate convergence rates on the residual norm of the underlying mapping for this general framework and then specialize it to obtain convergence guarantees for specific instances, where $k$ denotes the iteration counter. We extend our approach to a class of anchored Tseng's forward-backward-forward splitting methods to obtain a broader class of algorithms for solving co-hypomonotone inclusions. Again, we analyze $\mathcal{O}(1/k)$ last-iterate convergence rates for this general scheme and specialize it to obtain convergence results for existing and new variants. We generalize and unify Nesterov's accelerated extra-gradient method to a new class of algorithms that covers existing schemes as special instances while generating new variants. For these schemes, we can prove $\mathcal{O}(1/k)$ last-iterate convergence rates for the residual norm under co-hypomonotonicity, covering a class of nonmonotone problems. We propose another novel class of Nesterov's accelerated extragradient methods to solve inclusions. Interestingly, these algorithms achieve both $\mathcal{O}(1/k)$ and $o(1/k)$ last-iterate convergence rates, and also the convergence of iterate sequences under co-hypomonotonicity and Lipschitz continuity. Finally, we provide a set of numerical experiments encompassing different scenarios to validate our algorithms and theoretical guarantees.
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